1. PHYS 172: Modern Mechanics Summer 2012
DEMO: large gyroscope, like one on the next slide
DEMO Small gyroscope if wish – can hang it on a string
DEMO: Bicycle wheel
Additional files:
Gyro-faster-one-loop.mpg
Gyro-fast-one-loop.mpg
Lecture 20 – Angular Momentum Quantization Read 11.8 – 11.11

2. Predicting Position with Rotation
F=mg
A light string is wrapped around disk of radius R and moment of inertia I that can freely spin around its fixed axis. The string is pulled with force F during time Δt. Assume that the disk was initially at rest (ωi=0)
1) What will be the angular speed f ?  R I
Solution: m
There is a demo that corresponds to this situation, with the mass attached.

3. Predicting Position with Rotation
A light string is wrapped around disk of radius R and moment of inertia I that can freely spin around its fixed axis. The string is pulled with force F during time Δt. Assume that the disk was initially at rest (ωi=0)
What will be the angular speed ωf ?
How far (Δx) will the end of string move?  R I
See also examples
in Section 11.8
Solution:
 changes linearly with time:
F=mg m
Why not use calculus?

4. Angular momentum quantization
Many elementary particles behave as if they posses
intrinsic rotational angular momentum
Electron can have translational (orbital around nucleus), and intrinsic rotational angular momenta
Strange but true: Angular momentum is quantized
Angular momentum quantum =
Whenever you measure a vector component of angular momentum you get either half-integer or integer multiple of
Orbital angular momentum comes in integer multiples, but intrinsic spin of “Fermions” (building blocks) is ½ unit of

5. Orbital Angular Momentum
Where is the orbital angular momentum in a hydrogen orbital? + px
i py
Electron "current"
circles around
the atom.
Adding px+ipy gives the doughnut, color as phase, and it rotates in time. So, this is the orbital where the electron is actually circulating around the atom.
Angular momentum is quantized because of the spherical symmetry of the problem – the electron must take standing 3D spherical waves
about the nucleus. Bohr's idea was close, and much easier to calculate.
= |L=1, Lz=1>
Quantized because
these are 3D standing
electron waves
around the nucleus.
See Atom in a Box

6. Bohr’s Atomic Model
Allowed radii:
This implies that only certain values of LA,trans,electron are allowed:
NOTE: Because K and U are functions of r and v, energy levels are quantized also.
Niels Bohr
1913: IDEA: Electron can only take orbits where its translational angular momentum is integer multiple of

7. Bohr Model
Consider an electron in circular orbit about a proton. What are the possible values of LA,trans,electron? A
Thus,
If any orbital radius r is allowed, LA,trans,electron can be anything.
However, only certain values of r are allowed . . .
Assume circular motion:

8. The Bohr model: allowed radii and energies
See derivation on page
This is 2K
Use EN = K+U and
Bohr model energy levels: )

9. The Bohr model: and photon emission

10. Particle spin Rotational angular momentum
Electron, muon, neutrino have spin 1/2 :
mesurements of a component of their angular momentum yields ±½ħ
Quarks have spin ½
Protons and neutrons (three quarks) have spin ½
Mesons: (quark+antiquark) have spin 0 or 1
Macroscopic objects: quantization of L is too small to notice!
Two lowest energy electrons in any atom have total angular momentum 0
Fermions: spin ½, Pauli exclusion principle
Bosons: integer spin
Cooper pairs: superconductivity
Rotational energies of molecules are quantized
Quantum mechanics: Lx, Ly, Lz can only be integer or half-integer multiple of ħ
Quantized values of
where l is integer or half-integer

11. Gyroscopic Stability Edmund Scientifics
In 1917, the Chandler Company of Indianapolis, Indiana,
created the "Chandler gyroscope,” a toy gyroscope with a pull string and pedestal.
It has been in continuous production ever since and is considered a classic American toy.
-- Wikipedia 11

12. Vectors have direction
Best Trick in the Book
Vectors have direction
and magnitude.
Vector Notation and the Momentum Principle:
causes changes
in the magnitude of
in the direction of
Use the chain rule
Blast from the Past
Section 5.5

13. Best Trick Not in the Book
Vectors have direction
and magnitude.
Vector Notation and the Angular Momentum Principle:
causes changes
in the magnitude of
in the direction of
Use the chain rule
DEMO:
1) Bicycle tire on the stand demonstrates tau_perp, and changes in direction of L.
The larger the L, the harder it is to change its direction. 2)
demonstrates changes in magnitude of L

14. Gyroscopes Precession Precession and nutation
SHOW DEMO: large gyroscope
Possible first demo – hang small gyro on string, tell that it tries to stay at the same angle but precesses – this is what we about to talk about
We will only solve simple case – precession only, rotation axis is in horizontal plane.
Precession
Precession and nutation

15. Gyroscopes A B M CLICKER: What is the direction of , A or B?
A) Left B) Right
CLICKER: What is the direction of ?
A) Down C) into the page
B) Up D) out of the page
After that DEMO: bicycle wheel. Ask student to rotate its axle in horizontal plane – he will have difficulties. Then suggest that he should instead apply force up or down on one of the ends – and the axle will turn in horizontal plane!
For rotating vector:

16. i>clicker A A B
In which of the two gyroscopes is the disk spinning faster?

17. Precession phenomena (see book)
Magnetic Resonance Imaging (MRI)
Precession of spin axes in astronomy
Tidal torques
Felix Bloch
NMR - nuclear magnetic resonance
Edward Mills Purcell
Independently discovered (1946)
Nobel Price (1952)
Nuclear Magnetic Resonance Imaging – the word “nuclear” was dropped officially about 15 years ago as people feared anything associated with world “nuclear”.
B.S.E.E. from Purdue electrical engineering
NMRI = MRI